Though I am a novice in this forum, I would like to take a shot at this.
Dude, I have to say this is a great question, as this really touches the psychology of the English language in the context of grammatical usage vs colloquial usage.
To me, whatever can be thought of as saying what ever in 2 cases.
1. Message without an emotion (literal meaning): what ever as in “whatever you do, do it right” or “Do whatever it takes”.
2. Message with an emotion: In movies, whatever is commonly used in completion of a sentence which carries a casual emotion to it. Like “I don't know, whatever”. Here the tagline of the script must be that the actor displays a feeling of neglect.
Whereas whatsoever is more formal. It is not archaic yet as far as I know. It carries the same meaning as whatever. To me, whatsoever cannot carry an emotion. It is more correct when used formally.
So, to answer your question, if the city's signpost said “dump no waste whatever”, then I would say it is wrong. It must have been just “dump no waste”.